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This paper concerns the third-order m-point boundary-value problem $$displaylines{ u'''(t)+f(t,u(t),u'(t),u''(t))=0 ,quad hbox{a.e. } tin (0,1), cr u(0)=u'(0)=0, quad u''(1)=sum _{i=1}^{m-2}k_{i}u''(xi_{i}), }$$ where $f:[0,1]imes mathbb{R}^{3}o mathbb{R}$ is $L_p$-Caratheodory, $1leq p<+infty$, $0=xi_0<xi _1<dots <xi _{m-2}<xi_{m-1}=1$, $k_iin mathbb{R}$ ($i=1,2,dots ,m-2$) and $sum_{i=1}^{m-2}k_i eq 1$. Some criteria for the existence of at least one solution are established by using the well-known Leray-Schauder Continuation Principle.